Let X/R be a smooth scheme over a ring R. Consider the category of locally free crystals of finite rank on the situs Crys(X/Wt(R)). We show that it is equivalent to the category of locally free -modules of finite rank endowed with a nilpotent, integrable de Rham-Witt connection. In the case where R is a perfect field this was shown by Etesse [E] and Bloch [Bl]. We use the result for a construction of the Gauß-Manin connection as a de Rham-Witt connection.

The aim of this paper is to construct (i) infinitely many families of nonrational ℚ-Fano varieties of arbitrary dimension ≥ 4 with at most quotient singularities, and (ii) twelve families of nonrational ℚ-Fano threefolds with at most terminal singularities among which two are new and the remaining ten give an alternate proof of nonrationality to known examples. These are constructed as weighted hypersurfaces with the reduction mod p method introduced by Kollár [10].

Let K be a finite Galois extension of an algebraic number field k with G = Gal (K/k), and M be a Galois extension of k containing K. We denote by resp. the genus field resp. the central class field of K with respect to M/k. By definition, the field is the composite of K and the maximal abelian extension over k contained in M. The field is the maximal Galois extension of k contained in M satisfying the condition that the Galois group over K is contained in the center of that over k. Then it is well known that Gal is isomorphic to a factor group of the Schur multiplicator H-3(G, Z), and is isomorphic to H-3(G, Z) when M is sufficiently large. In this case we call M abundant for K/k (See Heider [3, § 4] and Miyake [6, Theorem 5]).

One branch of the research of submanifolds was introduced by Chen in terms of type in [2]. Type of a submanifold makes clear how the eigenspace decomposition of the Laplacian (of the ambient space) preserve after restricted to the submanifold.

Thanks to K. Heegner [He], A. Baker [Ba] and H. Stark [S], we know that there are nine imaginary quadratic fields of class number one. Gauss conjectured that there are infinitely many real quadratic fields of class number one, but the conjecture is still open.

Let be a quadratic field of discriminant d. We say that d is a prime discriminant if d is divisible by exactly one rational prime. It is classically known that the prime discriminants are given by

(p an odd prime).

Two vanishing theorems for harmonic map and L2 harmonic 1-form on complete noncompact manifolds are proved under certain geometric assumptions, which generalize results of [13], [15], [18], [19], and [20]. As applications, we improve some main results in [2], [4], [6], [9], [12], [20], [22], [24], and [25].

The concept of regular sequence of elements of a ring A (first introduced by Serre under the name of A-sequence [2]), has far-reaching uses in the theory of local rings and in algebraic geometry. It seems, however, that it loses much of its importance when A is not a noetherian ring, and in that case, it probably should be superseded by the concept of quasi-regular sequence [1].

In the present paper, it is proved that the K0-group of a Toeplitz algebra on any strongly pseudoconvex domain is always isomorphic to the K0-group of the relative continuous function algebra, and is thus isomorphic to the topological K0-group of the boundary of the relative domain. Further there exists a ring isomorphism between the K0-groups of Toeplitz algebras and the Chern classes of the relative boundaries of strongly pseudoconvex domains. As applications of our main result, K-groups of Toeplitz algebras on some special strongly pseudoconvex domains are computed. Our results show that the Toeplitz algebras on strongly pseudoconvex domains have rich structures, which deeply depend on the topological structures of relative domains. In addition, the first cohomology groups of Toeplitz algebras are also computed.

Let X(A, ω), ω∈Ω, be Wiener process on the probability space (Ω, B, P) depending on a point A of an N dimensional Euclidean space EN.

Let G be a connected linear algebraic group and X an algebraic variety, both defined over Q, the field of rational numbers. Suppose that G acts on X transitively and the action is defined over Q. Suppose that the set of rational points X(Q) is non-empty. Choosing x ∈ X(Q) allows us to identify G/Gx and X as varieties over Q, there Gx is the stabilizer of x.

Let Ω be a bounded pseudoconvex domain in Cn. We give sufficient conditions for the Bergman metric to go to infinity uniformly at some boundary point, which is stated by the existence of a Hölder continuous plurisubharmonic peak function at this point or the verification of property (P) (in the sense of Coman) which is characterized by the pluricomplex Green function.

One of the interesting problems in the potential theory for a function-kernel is to investigate the relation between the domination principle and the balayage principle.

Let M, N be positive definite quadratic lattices over Z with rank(M) = m and rank(N) = n respectively. When there is an isometry from M to N, we say that M is represented by N (even in the local cases). In the following, we assume that the localization Mp is represented by Np for every prime p. Let us consider the following assertion Am,n(N):

Am,n(N): There exists a constant c(N) dependent only on N so that M is represented by N if min(M) > c(N), where min(M) denotes the least positive number represented by M.

Consider the positive and twice continuously differentiable solutions u of the heat equation

in an open t-strip Ω = Rn×(0,T) for some T>0, where Rn is the n-dimensional Euclidean space.

In this note, we prove a theorem of Fatou type on u and, as its application, the uniqueness theorem for the Cauchy problem of ( 1 ).

Let Π be a finite group and denote by MΠ the class of finitely generated Z-free ZΠ-modules. In [2] we defined a certain equivalence relation on MΠ and constructed the abelian semigroup T(Π), which was studied in [3] (see [1] and [5], too).

Differential polynomial rings over a universal field and localized twisted polynomial rings over a separably closed field of non-zero characteristic twisted by the Frobenius endomorphism were the first domains not divisions rings that were shown to have every simple module injective (see [C] and [C-J]). By modifying the separably closed condition for the polynomial rings twisted by the Frobenius, the conditions of every simple being injective and only a single isomorphism class of simple modules were shown to be independent (see [O]). In this paper we continue the investigation of injective cyclic modules over twisted polynomial rings with coefficients in a commutative field.

Let L be a finite Galois extension of a number field K. Let G:= Gal(L/K). Let z1,…, zN ∊ L* \ {1} and let m1 …, mN ∊ ℚl. Let us assume that the linear combination of l-adic polylogarithms (constructed in some given way) is a cocycle on GL and that the formal sum is G-invariant. Then we show that cn determines a unique cocycle sn on GK. We also prove a weak version of Zagier conjecture for l-adic dilogarithm. Finally we show that if c2 is “motivic” (m1,…, mN ∊ ℚ) then s2 is also “motivic”.

The white noise calculus was initiated by T. Hida in 1970 in his Princeton University Mathematical Notes [3]. Recent development of the theory shows that the Laplacian plays an essential role in the analysis in question. Indeed, several kinds of Laplacians should be introduced depending on the choice of the class of white noise functionals to be analysed, as can be seen in [4], [13], [18] and so forth. Among others, we should like to emphasize the importance of the infinite dimensional Laplace-Beltrami operator, Volterra’s Laplacian and Lévy’s Laplacian (See [13], [18] and [20]).

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.